Finite Group Databasebeta

← #6487666

Subgroups — #6487666

Order 512.

2500 conjugacy classes of subgroups.

The subgroup lattice has 632 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#3101C211yesmaximal, characteristic, complemented
2C2×#3100C211yesmaximal, characteristic
3#35180C211yesmaximal, characteristic, complemented
4C2×D4×C4⋊C4C211yesmaximal, characteristic, complemented
5#32908C214yesmaximal, complemented
6#29760C214yesmaximal, complemented
7#24816C214yesmaximal, complemented
8#24812C214yesmaximal, complemented
9#24731C214yesmaximal, complemented
10#24727C214yesmaximal, complemented
11#21951C211yesmaximal, characteristic, complemented
12#21860C211yesmaximal, characteristic, complemented
13#21859C211yesmaximal, characteristic, complemented
14C2×#533C2211yescharacteristic
15C2×#532C2211yescharacteristic
16C22×C4×D4C2211yescharacteristic
17C22×C4×D4C2212yes—
18C23×C4⋊C4—21no—
19#3101—28no—
20#3100—28no—
21C2×#483C2212yes—
22C2×#480C2212yes—
23C2×#476C2212yes—
24C2×#474C2212yes—
25C2×#463C2218yes—
26C2×#461C2212yes—
27C2×#459C2212yes—
28C2×#450C2212yes—
29C2×#447C2212yes—
30C2×C4×SD16C2214yes—
31C2×C4×D8C2214yes—
32C2×#431C2211yescharacteristic
33C2×#431C2211yescharacteristic
34#2850C2214yes—
35#2639—22no—
36#2592—22no—
37#2568C2212yescomplemented
38#2563C2212yescomplemented
39#2562—22no—
40#2548C2212yescomplemented
41#2547C2212yescomplemented
42D4×C4⋊C4—22no—
43D4×C4⋊C4—22no—
44D4×C4⋊C4C2214yes—
45D4×C4⋊C4C2214yes—
46#2311—21no—
47#2311—21no—
48C2×#389C2212yes—
49C2×#385—22no—
50C2×#384C2211yescharacteristic
51C2×#380—21no—
52C2×C4×C4⋊C4C2211yescharacteristic
53#2085C2218yescomplemented
54#2068C2218yescomplemented
55#2065C2218yescomplemented
56#2042C2218yescomplemented
57#1990C2212yescomplemented
58#1990C2212yescomplemented
59#1957C2212yescomplemented
60#1956C2212yescomplemented
61#1939C2214yescomplemented
62#1939C2214yescomplemented
63#1879C2218yes—
64#1877C2218yes—
65#1874C2214yes—
66#1872C2214yes—
67#1804C2218yes—
68#1803C2214yescomplemented
69#1802C2214yescomplemented
70#1797C2214yes—
71C22×#41—21no—
72C23×D4C2311yescharacteristic
73C24×C4—21no—
74C24×C4—22no—
75C22×SD16C2314yes—
76C22×D8C2314yes—
77#533—24no—
78#533—24no—
79#532—24no—
80#532—24no—
81C2×C4⋊Q8C2311yescharacteristic
82C2×C4⋊Q8C2311yescharacteristic
83C2×C4⋊Q8C2311yescharacteristic
84C2×#127C2311yescharacteristic
85C2×#125—21no—
86C2×#124—21no—
87C2×#124C2314yes—
88C2×C22⋊Q8—21no—
89C2×C22⋊Q8C2312yes—
90C2×#121—21no—
91C2×#121C2312yes—
92C2×C4×Q8—21no—
93C2×C4×Q8C2312yes—
94C2×C4×D4—21no—
95C2×C4×D4—21no—
96C2×C4×D4—21no—
97C2×C4×D4—24no—
98C2×C4×D4—24no—
99C2×C4×D4—48no—
100C2×C4×D4D418yes—
101C2×C4×D4C2312yes—
102C2×C4×D4C2318yes—
103C2×C4×D4C2312yes—
104C2×C4×D4C2312yes—
105C2×#117—21no—
106C22×C4⋊C4—21no—
107C22×C4⋊C4—22no—
108C22×C4⋊C4—21no—
109C22×C4⋊C4—24no—
110C22×C4⋊C4—24no—
111C22×C4⋊C4—48no—
112C22×C4⋊C4D411yescharacteristic
113C22×C4⋊C4D411yescharacteristic
114C22×C4⋊C4C2311yescharacteristic
115C22×C4⋊C4C2312yes—
116C22×C22⋊C4—21no—
117C22×C22⋊C4—22no—
118C22×C42C2312yes—
119C22×C42C2311yescharacteristic
120#483—28no—
121#480—28no—
122#476—28no—
123#474—28no—
124#463—416no—
125#461—28no—
126#459—28no—
127#450—28no—
128#447—28no—
129C4×SD16—216no—
130C4×D8—216no—
131#431—24no—
132#431—24no—
133C2×#107C2314yes—
134C2×#106C2314yes—
135C2×C4⋊C8C2312yes—
136C2×C4⋊C8C2312yes—
137C2×C4⋊C8C2318yes—
138C2×#103C2314yes—
139C2×#103C2318yes—
140C2×#102C2314yes—
141C2×#102C2314yes—
142C2×#102C2314yes—
143C2×#102C2318yes—
144C2×C22⋊C8C2312yes—
145C2×C22⋊C8C2312yes—
146C2×C4×C8C2312yes—
147C2×C4×C8C2312yes—
148#400—24no—
149#397—24no—
150#394—24no—
151#393—24no—
152#391—22no—
153#391—22no—
154#391C2314yes—
155#390—22no—
156#390—22no—
157#390C2314yes—
158#389—24no—
159#389C2318yes—
160#388—24no—
161#388—24no—
162#386—24no—
163#386—24no—
164#385—28no—
165#385—28no—
166#384—21no—
167#384—21no—
168#384C2312yes—
169#384C2312yes—
170#382C2314yes—
171#380—22no—
172#380—22no—
173#380—22no—
174#380—22no—
175#380—24no—
176#380—24no—
177C4×C4⋊C4D412yes—
178C4×C4⋊C4D412yes—
179C4×C4⋊C4C2312yes—
180C4×C4⋊C4C2312yes—
181C2×#95—22no—
182C2×#95D412yes—
183#340C2318yes—
184#340C2318yes—
185#336C2318yes—
186C25—21no—
187C2×#41—44no—
188C2×#41—44no—
189C2×#41—44no—
190C22×Q8—21no—
191C22×Q8—21no—
192C22×Q8C2412yes—
193C22×D4—24no—
194C22×D4—24no—
195C22×D4—22no—
196C22×D4—21no—
197C22×D4—21no—
198C22×D4—21no—
199C22×D4—22no—
200C22×D4C2×D414yes—
201C22×D4C2412yes—
202C22×D4C2412yes—
203C23×C4—21no—
204C23×C4—28no—
205C23×C4—22no—
206C23×C4—22no—
207C23×C4—22no—
208C23×C4—22no—
209C23×C4—28no—
210C23×C4—48no—
211C23×C4—44no—
212C23×C4—44no—
213C23×C4—48no—
214C23×C4—84no—
215C23×C4C2×D412yes—
216C23×C4C2×D412yes—
217C23×C4C2×D411yescharacteristic
218C23×C4C2×D411yescharacteristic
219C23×C4C2411yescharacteristic
220C2×SD16—28no—
221C2×SD16—28no—
222C2×SD16—28no—
223C2×D8—28no—
224C2×D8—28no—
225C2×D8—28no—
226C22×C8C2414yes—
227C22×C8C2414yes—
228C4⋊Q8—22no—
229C4⋊Q8—22no—
230C4⋊Q8—22no—
231C4⋊Q8—22no—
232C4⋊Q8—22no—
233C4⋊Q8—22no—
234#127—22no—
235#127—22no—
236#125—44no—
237#124—48no—
238#124—44no—
239C22⋊Q8—24no—
240C22⋊Q8—24no—
241C22⋊Q8—44no—
242#121—24no—
243#121—24no—
244#121—44no—
245C4×Q8—24no—
246C4×Q8—24no—
247C4×Q8—44no—
248C4×D4—216no—
249C4×D4—28no—
250C4×D4—28no—
251C4×D4—24no—
252C4×D4—24no—
253C4×D4—24no—
254C4×D4—24no—
255C4×D4—48no—
256C4×D4—416no—
257C4×D4—44no—
258C4×D4C2×D418yes—
259C4×D4C2×D418yes—
260#117—44no—
261C2×C4⋊C4—24no—
262C2×C4⋊C4—24no—
263C2×C4⋊C4—21no—
264C2×C4⋊C4—22no—
265C2×C4⋊C4—22no—
266C2×C4⋊C4—22no—
267C2×C4⋊C4—28no—
268C2×C4⋊C4—22no—
269C2×C4⋊C4—21no—
270C2×C4⋊C4—21no—
271C2×C4⋊C4—21no—
272C2×C4⋊C4—21no—
273C2×C4⋊C4—22no—
274C2×C4⋊C4—22no—
275C2×C4⋊C4—22no—
276C2×C4⋊C4—22no—
277C2×C4⋊C4—44no—
278C2×C4⋊C4—416no—
279C2×C4⋊C4—416no—
280C2×C4⋊C4—48no—
281C2×C4⋊C4—44no—
282C2×C4⋊C4—82no—
283C2×C4⋊C4C2×D412yes—
284C2×C4⋊C4C2×D412yes—
285C2×C4⋊C4C2×D412yes—
286C2×C4⋊C4C2×D412yes—
287C2×C4⋊C4C2×D414yes—
288C2×C4⋊C4C2414yes—
289C2×C4⋊C4C2412yes—
290C2×C4⋊C4C2412yes—
291C2×C4⋊C4C2412yes—
292C2×C22⋊C4—28no—
293C2×C22⋊C4—22no—
294C2×C22⋊C4—22no—
295C2×C22⋊C4—22no—
296C2×C22⋊C4—28no—
297C2×C22⋊C4—22no—
298C2×C22⋊C4—22no—
299C2×C22⋊C4—24no—
300C2×C22⋊C4—24no—
301C2×C22⋊C4—24no—
302C2×C22⋊C4—24no—
303C2×C22⋊C4—48no—
304C2×C42—22no—
305C2×C42—44no—
306C2×C42C2×D414yes—
307C2×C42C2×D411yescharacteristic
308C2×C42C2×D411yescharacteristic
309C2×C42C2×D414yes—
310C2×C42#4114yes—
311C2×C42C2414yes—
312C2×C42C2411yescharacteristic
313C2×C42C2411yescharacteristic
314#107—28no—
315#106—28no—
316C4⋊C8—24no—
317C4⋊C8—24no—
318C4⋊C8—48no—
319#103—28no—
320#103—48no—
321#102—28no—
322#102—28no—
323#102—28no—
324#102—48no—
325C22⋊C8—24no—
326C22⋊C8—24no—
327C4×C8—24no—
328C4×C8—24no—
329#95—28no—
330#95—24no—
331#95—24no—
332#95—24no—
333#95—24no—
334#95C2×D414yes—
335#95C2×D414yes—
336C24—24no—
337C24—24no—
338C24—24no—
339C24—22no—
340C24—22no—
341C24—42no—
342C24—42no—
343C24—42no—
344C24—42no—
345C24C22×D411yescharacteristic
346#41—816no—
347C2×Q8—22no—
348C2×Q8—22no—
349C2×Q8—24no—
350C2×Q8—24no—
351C2×Q8—42no—
352C2×Q8—42no—
353C2×Q8—42no—
354C2×Q8—42no—
355C2×Q8—81no—
356C2×Q8—81no—
357C2×D4—24no—
358C2×D4—24no—
359C2×D4—216no—
360C2×D4—216no—
361C2×D4—24no—
362C2×D4—22no—
363C2×D4—22no—
364C2×D4—22no—
365C2×D4—22no—
366C2×D4—24no—
367C2×D4—22no—
368C2×D4—24no—
369C2×D4—44no—
370C2×D4—44no—
371C2×D4—44no—
372C2×D4—42no—
373C2×D4—42no—
374C2×D4—42no—
375C2×D4—42no—
376C2×D4—81no—
377C2×D4—81no—
378C2×D4C22×D414yes—
379C2×D4H(5,𝔽2)14yes—
380C2×D42−1+414yes—
381C22×C4—22no—
382C22×C4—22no—
383C22×C4—24no—
384C22×C4—24no—
385C22×C4—22no—
386C22×C4—24no—
387C22×C4—28no—
388C22×C4—28no—
389C22×C4—22no—
390C22×C4—22no—
391C22×C4—22no—
392C22×C4—24no—
393C22×C4—42no—
394C22×C4—41no—
395C22×C4—42no—
396C22×C4—41no—
397C22×C4—42no—
398C22×C4—42no—
399C22×C4—416no—
400C22×C4—416no—
401C22×C4—42no—
402C22×C4—42no—
403C22×C4—44no—
404C22×C4—44no—
405C22×C4—44no—
406C22×C4—44no—
407C22×C4—432no—
408C22×C4—44no—
409C22×C4—44no—
410C22×C4—44no—
411C22×C4—44no—
412C22×C4—44no—
413C22×C4—44no—
414C22×C4—88no—
415C22×C4—832no—
416C22×C4#12114yes—
417C22×C4#12114yes—
418C22×C4#13514yes—
419C22×C4AGL(1,ℤ/8)14yes—
420C22×C4C22×D414yes—
421C22×C4C22×D414yes—
422C22×C4C22×D411yescharacteristic
423C22×C4C22×D411yescharacteristic
424C22×C4C22×D411yescharacteristic
425C22×C4C22×D412yes—
426C22×C4C22×D411yescharacteristic
427C22×C4C22×D411yescharacteristic
428C22×C4C2×#4112yes—
429C22×C4C2511yescharacteristic
430SD16—416no—
431D8—416no—
432C2×C8—24no—
433C2×C8—24no—
434C2×C8—24no—
435C2×C8—24no—
436C2×C8—24no—
437C2×C8—24no—
438C4⋊C4—24no—
439C4⋊C4—24no—
440C4⋊C4—24no—
441C4⋊C4—24no—
442C4⋊C4—24no—
443C4⋊C4—22no—
444C4⋊C4—22no—
445C4⋊C4—22no—
446C4⋊C4—22no—
447C4⋊C4—22no—
448C4⋊C4—28no—
449C4⋊C4—416no—
450C4⋊C4—44no—
451C4⋊C4—42no—
452C4⋊C4—42no—
453C4⋊C4—42no—
454C4⋊C4—44no—
455C4⋊C4—44no—
456C4⋊C4—42no—
457C4⋊C4—44no—
458C4⋊C4—44no—
459C4⋊C4—82no—
460C4⋊C4—82no—
461C4⋊C4C22×D414yes—
462C22⋊C4—28no—
463C22⋊C4—28no—
464C22⋊C4—416no—
465C22⋊C4—44no—
466C22⋊C4—44no—
467C22⋊C4—44no—
468C22⋊C4—84no—
469C22⋊C4—84no—
470C22⋊C4—84no—
471C22⋊C4—84no—
472C42—28no—
473C42—44no—
474C42—44no—
475C42#12114yes—
476C42#12114yes—
477C42C22×D412yes—
478C42C22×D412yes—
479C42C2×#4112yes—
480C42C2×#4112yes—
481C23—21no—
482C23—21no—
483C23—21no—
484C23—21no—
485C23—216no—
486C23—24no—
487C23—24no—
488C23—24no—
489C23—48no—
490C23—48no—
491C23—48no—
492C23—48no—
493C23—48no—
494C23—42no—
495C23—42no—
496C23—48no—
497C23—42no—
498C23—42no—
499C23—42no—
500C23—42no—
501C23—42no—
502C23—42no—
503C23—42no—
504C23—42no—
505C23—42no—
506C23C2×#12112yes—
507C23C2×#13512yes—
508C23C2×AGL(1,ℤ/8)12yes—
509C23C23×D411yescharacteristic
510Q8—44no—
511Q8—44no—
512Q8—84no—
513Q8—84no—
514D4—216no—
515D4—24no—
516D4—24no—
517D4—24no—
518D4—44no—
519D4—44no—
520D4—84no—
521D4—88no—
522D4—84no—
523D4#53718yes—
524C2×C4—21no—
525C2×C4—21no—
526C2×C4—21no—
527C2×C4—21no—
528C2×C4—21no—
529C2×C4—21no—
530C2×C4—24no—
531C2×C4—24no—
532C2×C4—24no—
533C2×C4—24no—
534C2×C4—24no—
535C2×C4—24no—
536C2×C4—24no—
537C2×C4—24no—
538C2×C4—24no—
539C2×C4—44no—
540C2×C4—48no—
541C2×C4—48no—
542C2×C4—42no—
543C2×C4—42no—
544C2×C4—416no—
545C2×C4—416no—
546C2×C4—42no—
547C2×C4—44no—
548C2×C4—42no—
549C2×C4—42no—
550C2×C4—42no—
551C2×C4—42no—
552C2×C4—44no—
553C2×C4—42no—
554C2×C4—42no—
555C2×C4—42no—
556C2×C4—44no—
557C2×C4—42no—
558C2×C4—84no—
559C2×C4—832no—
560C2×C4—88no—
561C2×C4—88no—
562C2×C4#463116yescomplemented
563C2×C4C2×#12112yes—
564C2×C4C2×#12112yes—
565C2×C4C2×#12112yes—
566C2×C4C2×#12112yes—
567C2×C4C2×#12112yes—
568C2×C4C2×#13512yes—
569C2×C4C2×#13512yes—
570C2×C4C2×AGL(1,ℤ/8)12yes—
571C2×C4C2×AGL(1,ℤ/8)12yes—
572C2×C4C23×D411yescharacteristic
573C2×C4C22×#4111yescharacteristic
574C2×C4C2×H(5,𝔽2)12yes—
575C2×C4C2×2−1+412yes—
576C8—44no—
577C8—44no—
578C22—22no—
579C22—22no—
580C22—22no—
581C22—22no—
582C22—22no—
583C22—22no—
584C22—28no—
585C22—432no—
586C22—44no—
587C22—44no—
588C22—44no—
589C22—44no—
590C22—44no—
591C22—44no—
592C22—82no—
593C22—82no—
594C22—82no—
595C22—82no—
596C22—82no—
597C22—82no—
598C22—82no—
599C22C2×#46314yes—
600C22C22×#12111yescharacteristic
601C22#348411yescharacteristic
602C22#348611yescharacteristic
603C22C22×#13511yescharacteristic
604C22C22×AGL(1,ℤ/8)11yescharacteristic
605C22#358511yescharacteristic
606C22#358611yescharacteristic
607C4—22no—
608C4—22no—
609C4—22no—
610C4—44no—
611C4—44no—
612C4—44no—
613C4—82no—
614C4—84no—
615C4—82no—
616C4—82no—
617C4C2×#46314yes—
618C4C2×#46314yes—
619C4C2×#53712yes—
620C4#349212yes—
621C2—24no—
622C2—48no—
623C2—162no—
624C2#3522311yescharacteristic
625C2C22×#46311yescharacteristic
626C2#6042511yescharacteristic
627C2#6156511yescharacteristic
628C2#6158211yescharacteristic
629C2#6181911yescharacteristic
630C2#6182711yescharacteristic
subgroupG/H
center, socleC23C23×D4
commutator, Frattini, cosocleC22×C4C25
prime radicalC2×#3100C2
LCSG ▷ C22×C4 ▷ C2 ▷ C1
UCSC1 ◁ C23 ◁ C2×C4×C4⋊C4 ◁ G
LECSG ▷ C22×C4 ▷ C2 ▷ C1

Complements

Splittings G = N ⋊ H with N normal; H ≅ G/N.

NH ≅ G/Ntype
1. C2×#3101C2semidirect
3. #35180C2semidirect
4. C2×D4×C4⋊C4C2semidirect
5. #32908C2semidirect
6. #29760C2semidirect
7. #24816C2semidirect
8. #24812C2semidirect
9. #24731C2semidirect
10. #24727C2semidirect
11. #21951C2semidirect
12. #21860C2semidirect
13. #21859C2semidirect
37. #2568C22semidirect
38. #2563C22semidirect
40. #2548C22semidirect
41. #2547C22semidirect
53. #2085C22semidirect
54. #2068C22semidirect
55. #2065C22semidirect
56. #2042C22semidirect
57. #1990C22semidirect
58. #1990C22semidirect
59. #1957C22semidirect
60. #1956C22semidirect
61. #1939C22semidirect
62. #1939C22semidirect
68. #1803C22semidirect
69. #1802C22semidirect
562. C2×C4#463semidirect

Maximal subgroups

By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.

subgroupindex
C2×#31012
C2×#31002
#351802
C2×D4×C4⋊C42
#329082
#297602
#248162
#248122
#247312
#247272
#219512
#218602
#218592